<p style='text-indent:20px;'>In this paper, we study the dynamics of a linear control system with given state feedback control law in the presence of fast periodic sampling at temporal frequency <inline-formula><tex-math id="M1">\begin{document}$ 1/\delta $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M2">\begin{document}$ 0 &lt; \delta \ll 1 $\end{document}</tex-math></inline-formula>), together with small white noise perturbations of size <inline-formula><tex-math id="M3">\begin{document}$ \varepsilon $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M4">\begin{document}$ 0&lt; \varepsilon \ll 1 $\end{document}</tex-math></inline-formula>) in the state dynamics. For the ensuing continuous-time stochastic process indexed by two small parameters <inline-formula><tex-math id="M5">\begin{document}$ \varepsilon,\delta $\end{document}</tex-math></inline-formula>, we obtain effective ordinary and stochastic differential equations describing the mean behavior and the typical fluctuations about the mean in the limit as <inline-formula><tex-math id="M6">\begin{document}$ \varepsilon,\delta \searrow 0 $\end{document}</tex-math></inline-formula>. The effective fluctuation process is found to vary, depending on whether <inline-formula><tex-math id="M7">\begin{document}$ \delta \searrow 0 $\end{document}</tex-math></inline-formula> faster than/at the same rate as/slower than <inline-formula><tex-math id="M8">\begin{document}$ \varepsilon \searrow 0 $\end{document}</tex-math></inline-formula>. The most interesting case is found to be the one where <inline-formula><tex-math id="M9">\begin{document}$ \delta, \varepsilon $\end{document}</tex-math></inline-formula> are comparable in size; here, the limiting stochastic differential equation for the fluctuations has both a diffusive term due to the small noise and an effective drift term which captures the cumulative effect of the fast sampling. In this regime, our results yield a time-inhomogeneous Markov process which provides a strong (pathwise) approximation of the original non-Markovian process, together with estimates on the ensuing error. A simple example involving an infinite time horizon linear quadratic regulation problem illustrates the results.</p>

中文翻译：

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具有小随机噪声和快速周期采样的线性控制动态系统的逼近

) 在状态动态中。对于随后由两个小参数索引的连续时间随机过程 <inline-formula><tex-math id="M5">\begin{document}$ \varepsilon,\delta $\end{document}</tex-math ></inline-formula>，我们获得了有效的常微分方程和随机微分方程，描述了极限中的平均行为和平均值的典型波动为 <inline-formula><tex-math id="M6">\begin{document }$ \varepsilon,\delta \searrow 0 $\end{document}</tex-math></inline-formula>。发现有效波动过程会有所不同，取决于 <inline-formula><tex-math id="M7">\begin{document}$ \delta \searrow 0 $\end{document}</tex-math> </inline-formula> 快于/以相同的速度/慢于 < 内联公式><tex-math id="M8">\begin{document}$ \varepsilon \searrow 0 $\end{document}</tex-math></inline-formula>。发现最有趣的情况是 <inline-formula><tex-math id="M9">\begin{document}$ \delta, \varepsilon $\end{document}</tex-math>< /inline-formula> 大小相当；在这里，波动的极限随机微分方程由于噪声小而具有扩散项和捕获快速采样的累积效应的有效漂移项。在这种情况下，我们的结果产生了一个时间不均匀的马尔可夫过程，它提供了原始非马尔可夫过程的强（路径）近似，以及对随后误差的估计。